This is in essence, the cheapest way to start a business. You have rightly not stated what the product or service offering is. So we can only offer general advice.

I work making businesses more effective, both in terms of IT and lean innovation practises. There is nothing to stop someone setting up a business and making the same thing or offering the same service as someone else, but faster and cheaper. It certainly isn't illegal as long as you don't violate a patent or other intellectual property in some way. The reality is you have to understand what IP exists (copyright, design-right, patent, registered trademarks...) and not every one of those applies to all services. Indeed, if a product or service is "obvious" to an expert in the industry, or it's public domain, patenting it can be considered indefensible anyway.

Most businesses who are second entrants off the back of a primary entrant who's idea is proving itself, have a much cheaper entry to the market and they can build on the lessons their first mover competitors have learned (or failed to learn). Doing what you describe can still be innovative. The classical example of the Toyota production system transformed the production of cars not just in Japan, but across the world. They couldn't patent the car, since that existed already and the production system they created could not defensibly be patented anyway. In any event, make sure your IP is protected once you do manage to do it.

If you'd like to discuss this further, feel free to book yourself a call. I'd be happy to help walk you through it.